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homework_1 - ELECTRODYNAMICS PROBLEM SET 1 due February 2nd...

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ELECTRODYNAMICS PROBLEM SET 1 due February 2nd, before class Problem 1.: tensor (anti)-symmetry Show that if A μν = A νμ is a symmetric tensor in one frame, it’ll be symmetric in any other frame. Is anti -symmetry ( A μν = - A νμ ) also frame independent? Show that if S μν is symmetric ( S μν = S νμ ) and A μν is antisymmetric ( A μν = - A νμ ) then A μν S μν = 0. Show that ∇ × ∇ f = 0 , . ∇ × ~a = 0 , (1) for sufficiently well-behaved f and ~a . Problem 2.: Most of vector/tensor analysis The index notation discussed in class is very useful, eve if all you want to do is to deal with vectors (not tensors) in 3D. But you need a little practice to use it efficiently. Instead of memorizing all those identities you find on the back of textbooks, you need only the fundamental relation ijk ilm = δ jl δ km - δ jm δ kl . (2) Show the relation above (hint: what else could the right-hand side be?). It is then easy to show ijk ijm = 2 δ km (3) Now you are ready to show c) Show that ~ A. ( ~ B × ~ C ) = ~ B. ( ~ C × ~ A ) = ~ C. ( ~ A × ~ B ) (4) ( ~ A.
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